Impact of elastic inhomogeneity on collective dynamical properties investigated by field theoretical description in real space

TL;DR

The paper addresses the origin of the Boson peak and related low-frequency vibrational anomalies in amorphous solids, challenging Debye-based interpretations. It adopts a real-space field-theoretic description of an inhomogeneous elastic medium and solves the Green's function problem numerically, enabling direct visualization of local DOS. The main findings show that elastic inhomogeneity causes selective scattering of short-wavelength modes, elevating the low-frequency DOS and shifting the Boson peak to ω_BP ∼ $\bar{\kappa}^{1/2}/(2L)$, where L is the correlation length; moreover, vibrational inhomogeneity emerges as highly excited spots in soft regions, visible in real space. These results demonstrate that a BP-like excess can arise without a flat dispersion and provide a concrete link between mesoscopic elasticity, real-space vibrational structure, and particle-level observations, offering a bridge between continuum theory and experiments.

Abstract

Interpreting the vibrational properties of amorphous solids beyond Debye's theory is challenging due to the presence of inhomogeneity on the mesoscopic scale. In this work, we model this inhomogeneity by real-space fluctuating elasticity with a spatially correlated distribution and calculate the dynamical properties using an exact real-space field theoretical approach. Our results clarify that the excess low-frequency density of states (DOS) originates from a selective scattering effect (stronger scattering of short wavelengths) induced by elastic inhomogeneity. The visualization of the local DOS in real space reveals the existence of anomalous modes, highly excited spots, at low frequencies. The findings regarding these highly excited spots and the selectivity of the correlation length were missed in previous perturbative field approaches in wave-vector space, and they align with recent progress from particle-level simulations and experiments. These results provide concrete insights into the low-frequency vibrational anomaly of amorphous solids from the perspective of simple elastic inhomogeneity.

Figures (5)

• Figure 1: Real space distribution of $\kappa (\boldsymbol{r})$ with fluctuation intensities of $0$ (homogeneous) a, $0.5$ (inhomogeneous) b, and $0.99$ (inhomogeneous) c. The mean value is $\bar{\kappa} = 1$, and the correlation length is $0.01$. Spectral functions $A(\omega, q)$ are shown in d, e, and f, corresponding to the inhomogeneity levels in a, b, and c, respectively. Cuts of the spectral function $A(\omega, q)$ at various wave vectors $q$ are shown in g, h, and i, for the same cases.
• Figure 2: a: The reduced DOS, $g(\omega)/\omega$, for different inhomogeneity intensities $0$, $0.5$, and $0.99$ (gray, darker gray, and black, respectively), computed using Eq. \ref{['dosq']}. The $\kappa$ distributions correspond to those in Fig. \ref{['figintensity']} a--c. The vertical dashed line indicates the Boson peak (BP) position around $\omega_{BP} \sim 40$. The gray-shaded high-frequency region ($\omega > 100$) corresponds to the Van Hove peak, where the dispersion relation deviates from linearity as seen in Fig. \ref{['figintensity']} d--f. b: The reduced half-peak width $\Gamma (q)$ of the spectral function, reflecting the flatness of $A(\omega, q)$ at different wave vectors $q$, as shown in Fig. \ref{['figintensity']} g--i.
• Figure 3: The reduced DOS, $g(\omega)/\omega$, for different correlation lengths $0.005N$, $0.01N$, and $0.015N$ (gray, darker gray, and black, respectively), a. The DOS is computed using Eq. \ref{['dosq']}, with an inhomogeneity intensity of $0.99$. The vertical dashed lines indicate the BP positions at $\omega_{BP} \sim 72, 40, 20$ for the respective correlation lengths.
• Figure 4: The local DOS $A_{\omega} (\boldsymbol{r})$ at different frequencies $\omega = 1, 10, 20, 30, 40, 50, 70, 90, 110$ (panels a to i). The local DOS is computed from Eqs. \ref{['greenreal']}, \ref{['realspectra']}, and \ref{['dosreal']}. The corresponding elastic modulus distribution $\kappa(\boldsymbol{r})$ is shown in Fig. \ref{['figkappa']}. The dashed white boxes highlight several (four) highly excited spots (regions of large local DOS) at low frequencies $10 < \omega < 50$. Their positions correspond to soft spots (relatively small $\kappa$) indicated by dashed white boxes and white crosses in Fig. \ref{['figkappa']}.
• Figure 5: Distribution of the elastic modulus $\kappa (\boldsymbol{r})$ used for calculating the local DOS shown in Fig. \ref{['figldos']}. The inhomogeneity intensity is $0.99$, and the correlation length is $L = 0.01 N$. White crosses within the dashed white box mark the positions of highly excited spots (large local DOS) at low frequencies $10 < \omega < 50$, as highlighted by the dashed white boxes in Fig. \ref{['figldos']}.